Authors:Qifa Lin, Runxin Wu, Lin LiAbstract: A two species commensal symbiosismodel with Holling type functional response and Allee effect on the second species takes the form $$\begin{array}{rcl}\di\frac{dx}{dt}&=&x\Big(a_1-b_1x+\di\frac{c_1y^p}{1+y^p}\Big),\\[4mm]\di\frac{dy}{dt}&=&y(a_2-b_2y)\di\frac{y}{u+y}\end{array}$$ is investigated, where $a_i, b_i, i=1,2$ $p$, $u$ and $c_1$ are all positive constants, $p\geq 1$. Local and global stability property of the equilibria is investigated. Our study indicates that the unique positive equilibrium is globally stable and the system always permanent. Our study shows that Allee effect has no influence on the final density of the species. However, numeric simulations show that the stronger the Allee effect, the longer the for the system to reach its stable steady-state solution.PubDate: 2018-03-16Issue No:Vol. 2018 (2018)

Authors:Lan Guijie, Fu Yingjie, Wei Chunjin, Zhang ShuwenAbstract: In this paper, we study the dynamics of the stochastic SI epidemic model for pest management concerning spraying pesticide and releasing natural enemies. Existence of a unique global positive solution is proved firstly. And we show that the positive solution to the stochastic system is stochastically bounded. Third, by using Khasminshii's method and Lyapunov function, we derive the sufficient conditions for the existence of the nontrivial stochastically positive T-periodic solution. Then, by comparison theorem for stochastic differential equation, the sufficient conditions for existence and global attraction of the boundary periodic solution are obtained. Finally, Numerical simulations are carried out to substantiate the analytical results.PubDate: 2018-03-15Issue No:Vol. 2018 (2018)

Authors:Qifa LinAbstract: A two species commensal symbiosismodel with non-monotonic functional response and non-selective harvesting in a partial closure takes the form $$\begin{array}{rcl}\di\frac{dx}{dt}&=&x\Big(a_1-b_1x+\di\frac{c_1y }{d_1+y^2}\Big)-q_1Emx,\\[4mm]\di\frac{dy}{dt}&=&y(a_2-b_2y)-q_2Emy\end{array}$$ is proposed and studied, where $a_i, b_i, q_i, i=1,2$ $c_1$, $E$, $m(0<m<1)$ and $d_1$ are all positive constants. Depending on the range of the parameter $m$, the system may be collapse, or partial survival, or the two species could be coexist in a stable state. We also show that if the system admits a unique positive equilibrium, then it is globally asymptotically stable. By introducing the harvesting term and the reserve area, the system exhibit rich dynamic behaviors. Our results generalize the main results of Chen and Wu (A commensal symbiosis model with non-monotonic functional response, Commun. Math. Biol. Neurosci. Vol 2017 (2017), Article ID 5)PubDate: 2018-02-21Issue No:Vol. 2018 (2018)

Authors:S. Y. Tchoumi, J. C. Kamgang, D. Tieudjo, G. SalletAbstract: We propose a model that can translate the dynamics of vector-borne diseases, for this model we compute the basic reproduction number and show that if $\mathcal{R}_0<\zeta<1$ the DFE is globally asymptotically stable. For $\mathcal {R}_0>1$ we prove the existence of a unique endemic equilibrium and if $\mathcal {R}_0 \leq 1$ the system can have one or two endemic equilibrium, we also show the existence of a backward bifurcation. By numerical simulations we illustrate with data on malaria all the results including existence, stability and bifurcation.PubDate: 2018-02-12Issue No:Vol. 2018 (2018)

Authors:I. Agmour, M. Bentounsi, N. Achtaich, Y. El FoutayeniAbstract: The present paper describes a prey-predator type fishery model with two predators in competition. The aim of the paper is to maximize the net economic revenue earn from the fishery through implementing the sustainable properties of the fishery to keep the ecological balance. The existence of the steady states and the stability of the interior equilibrium point is studied using Routh Hurwitz criterion. The problem of determining the fishing effort that maximizes the net economic revenue of each fisherman results in a Generalized Nash Equilibrium Problem. More precisely, we are interested in equilibrium of mathematical game given by the situation where all fishermen try to optimize their strategies according to the strategies of all other fishermen. The importance of marine reserve is analyzed through the obtained results of the numerical simulations of proposed model system. The results depict that reserves will be most effective when the coefficient of catchability decreases.PubDate: 2018-01-22Issue No:Vol. 2018 (2018)

Authors:Wenwen Zhang, Shaokun Lu, Yongzhen PeiAbstract: This paper proposed a control-strategies for nodes to control the spread of an epidemic outbreak in arbitrary directed graphs by optimally allocating their resources throughout the network. Epidemic propagation is well modeled as a networked version of the Susceptible-Exposed-Infected-Susceptible (SEIS) epidemic process. Using the Kolmogorov forward equations and mean-field approximation, we present a mean-field model to describe the spreading dynamics and prove the existence of a necessary and sufficient condition for global exponential stability. Based on this stability condition, we can derive another condition to control the spread of an epidemic outbreak in terms of the eigenvalues of a matrix that depends on the network structure and the parameters of the model. According to different control purposes and conditions, two types of control-theoretic decision can be considered: 1)given a fixed budget, find the optimal resource allocation to achieve the highest level of containment, 2)given a decay rate of epidemic, find the minimum cost to control the spreading process at a desired decay rate. A geometric program can be formulated to solve the optimal problems and the existence of solutions is also proved. Numerical simulations can illustrated our results.PubDate: 2018-01-05Issue No:Vol. 2018 (2018)